Extruding a circular shape from a non-circular object
Depending on your mesh, you will have to select a number of faces to use as a circular entrance to the hole. The density of your example image suggests to use 2x2 (4) faces.


Select the faces and remove them. X
Position the cursor at the center of the desired hole and set the pivo point to cursor mode.
Select the vertices and make them to sphere &#8679; Shift&#9095; AltS > 1 > &#9166; Enter. 
Smooth them with the loop tools space option.
Add a circle with 8 vertices and bridge the remaining geometry. You know have a perfectly circular piece of geometry which you can extrude inwards.




The resulting topology